On the Stability of Modified Patankar Methods

Patankar schemes have attracted increasing interest in recent years because they preserve the positivity of the analytical solution of a production-destruction system (PDS) irrespective of the chosen time step size. Although they are now of great interest, for a long time it was not clear what stability properties such schemes have. Recently a new stability approach based on Lyapunov stability with an extension of the center manifold theorem has been proposed to study the stability properties of positivity preserving time integrators. In this work, we study the stability properties of the classical modified Patankar--Runge--Kutta schemes (MPRK) and the modified Patankar Deferred Correction (MPDeC) approaches. We prove that most of the considered MPRK schemes are stable for any time step size and compute the stability function of MPDeC. We investigate its properties numerically revealing that also most MPDeC are stable irrespective of the chosen time step size. Finally, we verify our theoretical results with numerical simulations.Comment: 34 pages, 14 Figure

Lyapunov Stability of First and Second Order GeCo and gBBKS Schemes

In this paper we investigate the stability properties of fixed points of the so-called gBBKS and GeCo methods, which belong to the class of non-standard schemes and preserve the positivity as well as all linear invariants of the underlying system of ordinary differential equations for any step size. The schemes are applied to general linear test equations and proven to be generated by $\mathcal C^1$-maps with locally Lipschitz continuous first derivatives. As a result, a recently developed stability theorem can be applied to investigate the Lyapunov stability of non-hyperbolic fixed points of the numerical method by analyzing the spectrum of the corresponding Jacobian of the generating map. In addition, if a fixed point is proven to be stable, the theorem guarantees the local convergence of the iterates towards it. In the case of first and second order gBBKS schemes the stability domain coincides with that of the underlying Runge--Kutta method. Furthermore, while the first order GeCo scheme converts steady states to stable fixed points for all step sizes and all linear test problems of finite size, the second order GeCo scheme has a bounded stability region for the considered test problems. Finally, all theoretical predictions from the stability analysis are validated numerically.Comment: 31 pages, 7 figure

ESQUEMA DEL TIPO MODIFIED PATANKAR RUNGE–KUTTA UTILIZANDO UNA PONDERACIÓN CONVEXA DE LOS PESOS PATANKAR WEIGHT DENOMINATORS

The following presents an introduction to the modified patankar rungekutta schemes, which are known as numeric schemes that solve systems of production destruction positives and conservatives. It will show the main properties of MPRK schemes, in particular it stands out the second order schemes of two stages known as MPRK22. Further, it would intro- duce a modification of the MPRK22 scheme, and show a new scheme of second order by completing a convex combination of the Patankar Weight Denominators (PWD).Este trabajo presenta una introducción a los esquemas del tipo Modified Patankar Runge-Kutta (MPRK), los esquemas numéricos MPRK resuelven sistemas de Producción Destrucción positivos y conservativos. Se presenta las principales propiedades de los esquemas MPRK y en particular se detalla el esquema de segundo orden de dos etapas denominado MPRK22, para luego introducir una modificación del esquema MPRK22 y presentar un nuevo esquema de segundo orden haciendo una combinación convexa de los pesos Patankar Weight Denominators (PWD). Los resultados son confirmados por experimentos numéricos considerando un sistema de ecuaciones diferenciales no lineal

On the Accuracy of Explicit Finite-Volume Schemes for Fluctuating Hydrodynamics

This paper describes the development and analysis of finite-volume methods for the Landau–Lifshitz Navier–Stokes (LLNS) equations and related stochastic partial differential equations in fluid dynamics. The LLNS equations incorporate thermal fluctuations into macroscopic hydrodynamics by the addition of white noise fluxes whose magnitudes are set by a fluctuation-dissipation relation. Originally derived for equilibrium fluctuations, the LLNS equations have also been shown to be accurate for nonequilibrium systems. Previous studies of numerical methods for the LLNS equations focused primarily on measuring variances and correlations computed at equilibrium and for selected nonequilibrium flows. In this paper, we introduce a more systematic approach based on studying discrete equilibrium structure factors for a broad class of explicit linear finite-volume schemes. This new approach provides a better characterization of the accuracy of a spatiotemporal discretization as a function of wavenumber and frequency, allowing us to distinguish between behavior at long wavelengths, where accuracy is a prime concern, and short wavelengths, where stability concerns are of greater importance. We use this analysis to develop a specialized third-order Runge–Kutta scheme that minimizes the temporal integration error in the discrete structure factor at long wavelengths for the one-dimensional linearized LLNS equations.Together with a novel method for discretizing the stochastic stress tensor in dimension larger than one, our improved temporal integrator yields a scheme for the three-dimensional equations that satisfies a discrete fluctuation-dissipation balance for small time steps and is also sufficiently accurate even for time steps close to the stability limit

A Low Mach Number Solver: Enhancing Applicability

In astrophysics and meteorology there exist numerous situations where flows exhibit small velocities compared to the sound speed. To overcome the stringent timestep restrictions posed by the predominantly used explicit methods for integration in time the Euler (or Navier-Stokes) equations are usually replaced by modified versions. In astrophysics this is nearly exclusively the anelastic approximation. Kwatra et al. have proposed a method with favourable time-step properties integrating the original equations (and thus allowing, for example, also the treatment of shocks). We describe the extension of the method to the Navier-Stokes and two-component equations. - However, when applying the extended method to problems in convection and double diffusive convection (semiconvection) we ran into numerical difficulties. We describe our procedure for stabilizing the method. We also investigate the behaviour of Kwatra et al.'s method for very low Mach numbers (down to Ma = 0.001) and point out its very favourable properties in this realm for situations where the explicit counterpart of this method returns absolutely unusable results. Furthermore, we show that the method strongly scales over 3 orders of magnitude of processor cores and is limited only by the specific network structure of the high performance computer we use.Comment: author's accepted version at Elsevier,JCP; 42 pages, 14 figure

Staggered Schemes for Fluctuating Hydrodynamics

We develop numerical schemes for solving the isothermal compressible and incompressible equations of fluctuating hydrodynamics on a grid with staggered momenta. We develop a second-order accurate spatial discretization of the diffusive, advective and stochastic fluxes that satisfies a discrete fluctuation-dissipation balance, and construct temporal discretizations that are at least second-order accurate in time deterministically and in a weak sense. Specifically, the methods reproduce the correct equilibrium covariances of the fluctuating fields to third (compressible) and second (incompressible) order in the time step, as we verify numerically. We apply our techniques to model recent experimental measurements of giant fluctuations in diffusively mixing fluids in a micro-gravity environment [A. Vailati et. al., Nature Communications 2:290, 2011]. Numerical results for the static spectrum of non-equilibrium concentration fluctuations are in excellent agreement between the compressible and incompressible simulations, and in good agreement with experimental results for all measured wavenumbers.Comment: Submitted. See also arXiv:0906.242

Homogeenisen seosmallin verifiointi vapaan nestepinnan ongelmaan

In this thesis, the applicability of the homogeneous mixture model of Finflo for the free surface problem is studied. The free surface problem is fundamental in marine hydrodynamics, and a special case in two phase flows. The work explores the basis of this type of modelling from mathematical and numerical viewpoint, and verifies the mixture model for the problem. The mathematical background of the problem is presented, together with the nature of it from the perspective of marine hydrodynamics. The bulk flow equations are usually averaged conditionally such that the governing equations of the multiphase model are formally the same as in the case of single phase flow. It can be shown that one additional equation suffices for the description of the segregated phases. Here, the convection equation of the void fraction is utilized. The void fraction equation is derived in conservative form based on the incompressibility constraint of the individual phases. The convection of the void fraction corresponds to the so-called Riemann problem. This is studied thoroughly by developing a two-dimensional solver for the comparison of some well-known schemes for the spatial discretisation of the convective quantity. This solver is applied to the convection of a discontinuous distribution of the void fraction. In addition, the so-called SUPERBEE limiter is implemented to the Finflo code for the extrapolation of the convective void fraction. The numerical solution of the Navier-Stokes equations for simulations of two phase flows is covered comprehensively. The code Yaffa, developed at the Aalto University, has a modern VOF model implemented, and for this reason, it is here used as a reference code. The solution algorithms, the computation of the convective quantities, the pressure correction stages as well as the treatment of the segregated phases in both of the codes are discussed in detail. The two phase flow over a submerged ground elevation is computed using the codes Finflo and Yaffa, and the forming free surface wave is compared to those found from the literature. The aim of this thesis is to get acquainted with the nature of the problem in conjunction with the specific methodology used to solve such flows. This is done in order to understand the requirements and possible modifications needed for the model when we wish to accurately predict ship flow phenomena that are not solvable using the traditional free surface tracking strategies. This way, the verification of the mixture model of Finflo is achieved.Tässä työssä tutkitaan Finflon homogeenisen seosmallin soveltuvuutta vapaan nestepinnan ongelmaan. Vapaan nestepinnan ongelma on keskeinen laivahydrodynamiikassa, ja samalla monifaasivirtauksien erikoistapaus. Työssä perehdytään tällaisen mallinnuksen perusteisiin matemaattisessa ja numeerisessa mielessä, ja verifioidaan samalla seosmallia tälle ongelmalle. Työssä esitetään ongelman matemaattinen tausta sekä sen luonne laivahydrodynamiikan kannalta. Virtausta kuvaavat yhtälöt yleensä keskiarvostetaan ehdollisesti se. käytettävän monifaasimallin perusyhtälöt ovat muodollisesti samat, kuin yksifaasisessakin tapauksessa. Voidaan osoittaa, että tässä tapauksessa erillisten faasien kuvaukseen riittää yksi lisäyhtälö, joksi työssä otetaan aukko-osuuden konvektioyhtälö. Aukko-osuusyhtälö johdetaan säilymismuodossa perustuen faasien kokoonpuristumattomuusoletukseen. Mainittu lisäyhtälö vastaa luonteeltaan konvektioyhtälön ns. Riemann-probleemaa, ja tätä käsitellään perusteellisesti. Työssä kehitetään kaksidimensioinen ratkaisija, jolla vertaillaan tunnettuja menetelmiä konvektoituvan suureen paikkadiskretoinnille soveltamalla sitä epäjatkuvan aukko-osuusjakauman konvektioprobleemalle. Lisäksi implementoidaan Finfloon ns. SUPERBEE-rajoitin konvektoituvan aukko-osuuden ekstrapolointiin. Työssä käsitellään kattavasti Navier-Stokes -yhtälöiden numeerista ratkaisua kaksifaasivirtausimulointimenetelmien kannalta. Referenssikoodiksi otetaan Aalto-yliopistossa kehitetty Yaffa, johon nykyaikainen VOF-malli on implementoitu. Muiden muassa koodien ratkaisualgoritmi, konvektoituvien suureiden laskenta, painekorjausvaihe sekä erottuneiden faasien käsittely kuvataan perusteellisesti. Finflo- ja Yaffa -koodeilla lasketaan kaksifaasivirtaus vedenalaisen kummun yli, ja syntynyttä aaltokuviota verrataan myös kirjallisuudesta löytyviin tuloksiin. Työn ajatuksena on tutustua vapaan nestepinnan ongelman luonteeseen yhdessä tällaisen yleisemmän ratkaisutavan kanssa. Tavoitteena on ymmärtää mallille asetettavia vaatimuksia sekä sitä, millaisia modifikaatioita siihen tulisi tehdä, kun esim. pyritään ennustamaan tarkasti sellaisia laivavirtauksiin liittyviä ilmiöitä, joihin perinteiset pintaa seuraavat mallit eivät pysty. Tällä tavalla saatiin Finflon seosmallin verifiointi aikaiseksi